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Non-archimedean analytic continuation of unobstructedness

Hang Yuan

TL;DR

The paper introduces a non-archimedean analytic framework for unobstructedness in Lagrangian Floer theory, defining a stronger notion called proper unobstructedness that tracks the vanishing of obstruction series in an affinoid algebra. It proves that proper unobstructedness is invariant under choices and that unobstructedness propagates along any connected family of Lagrangians via analytic continuation in the non-archimedean setting, aided by Fukaya’s trick. The main contributions include a precise wall-crossing theory for superpotentials and obstruction data, a category of A_\infty algebras with topological labels (the UD category), and a rigorous non-archimedean perspective that provides robust continuation results beyond the Maurer–Cartan picture. These results have implications for SYZ mirror symmetry, wall-crossing phenomena, and the global unobstructedness of fibers in Lagrangian fibrations, with concrete applications to monotone tori and blow-up toric geometries. Overall, the work offers a novel analytic framework for understanding unobstructedness and its continuation across Lagrangian families, linking family Floer ideas with non-archimedean geometry.

Abstract

The Floer cohomology and the Fukaya category are not defined in general. Indeed, while the issue of obstructions can be theoretically addressed by introducing bounding cochains, the actual existence of even one such bounding cochain is usually unknown. This paper aims to deal with the problem. We study certain non-archimedean analytic structure that enriches the Maurer-Cartan theory of bounding cochains. Using this rigid structure and family Floer techniques, we prove that within a connected family of graded Lagrangian submanifolds, if any one Lagrangian is unobstructed (in a slightly stronger sense), then all remaining Lagrangians are automatically unobstructed.

Non-archimedean analytic continuation of unobstructedness

TL;DR

The paper introduces a non-archimedean analytic framework for unobstructedness in Lagrangian Floer theory, defining a stronger notion called proper unobstructedness that tracks the vanishing of obstruction series in an affinoid algebra. It proves that proper unobstructedness is invariant under choices and that unobstructedness propagates along any connected family of Lagrangians via analytic continuation in the non-archimedean setting, aided by Fukaya’s trick. The main contributions include a precise wall-crossing theory for superpotentials and obstruction data, a category of A_\infty algebras with topological labels (the UD category), and a rigorous non-archimedean perspective that provides robust continuation results beyond the Maurer–Cartan picture. These results have implications for SYZ mirror symmetry, wall-crossing phenomena, and the global unobstructedness of fibers in Lagrangian fibrations, with concrete applications to monotone tori and blow-up toric geometries. Overall, the work offers a novel analytic framework for understanding unobstructedness and its continuation across Lagrangian families, linking family Floer ideas with non-archimedean geometry.

Abstract

The Floer cohomology and the Fukaya category are not defined in general. Indeed, while the issue of obstructions can be theoretically addressed by introducing bounding cochains, the actual existence of even one such bounding cochain is usually unknown. This paper aims to deal with the problem. We study certain non-archimedean analytic structure that enriches the Maurer-Cartan theory of bounding cochains. Using this rigid structure and family Floer techniques, we prove that within a connected family of graded Lagrangian submanifolds, if any one Lagrangian is unobstructed (in a slightly stronger sense), then all remaining Lagrangians are automatically unobstructed.
Paper Structure (39 sections, 35 theorems, 76 equations, 1 figure)

This paper contains 39 sections, 35 theorems, 76 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main result
  3. Applications
  4. Unobstructedness beyond the Maurer-Cartan picture
  5. Traditional unobstructedness
  6. Proper unobstructedness
  7. Non-archimedean analytic continuation
  8. Inspiration from family Floer mirror symmetry
  9. Related work
  10. $A_\infty$ algebras with topological labels
  11. Gappedness condition
  12. Retrieval of filtered $A_\infty$ algebras
  13. Pseudo-isotopies of $A_\infty$ algebras
  14. Category $\mathscr{UD}$
  15. Homological perturbation and harmonic contractions
  16. ...and 24 more sections

Key Result

Theorem 1.1

If $L_{s_0}$ is properly unobstructed, then all $L_s$ are properly unobstructed.

Figures (1)

  • Figure :

Theorems & Definitions (69)

  • Theorem 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Theorem 1.5: Theorem \ref{['proper_unobstructedness_invariance_thm']}
  • Proposition 1.6
  • Proposition 1.7
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • ...and 59 more